Let $\pi$ is probability measures on $(X,\mathscr{B}\left(X\right))$, where $\mathscr{B}\left(X\right)$ is borel $\sigma\text{-algebra}$ on $X$. Consider the set of all continuity sets in $\mathscr{B}\left(X\right)$. It means that consider set of all $B\in\mathscr{B}\left(X\right)$ such, that $\pi(\partial B)=0$. Morover consider that $f:X\to X$ is borel measurable function, it means that $\forall B\in\mathscr{B}\left(X\right); f^{-1}(B)=\{x\in X; f(x)\in B\}\in \mathscr{B}\left(X\right)$.
I would like to know if it is true that the preimage of continuity set in $f$ is continuity set. In other words, if the set $A=f^{-1}(B)$, where $B$ is continuity set, is also continuity set.
I have no idea how to start. Any help will be appreciated. Thank you very much.
Let $X=\mathbb R$ and $f(x)=1$ for $x \geq 0, 0$ for $x <0$. Let $\pi$ be any probability measure on the Borel sets of $\mathbb R$ such that $\pi (\{1\})=0$ and $\pi (\{0\}) \neq 0$. (For example, $\pi$ could be the delta measure at $0$). Then $\{1\}$ is a continuity set but $f^{-1}(\{1\}) =[0,\infty)$ is not, since the boundary of $[0,\infty)$ is $\{0\}$.
Even continuity of $f$ is not enough. Take $f(x)=x+1$ with the measure $\pi$ described above.